ALR 2014 Caual Modelling Workshop

Transcription

1 ALR 2014 Caual Modelling Workshop Example 1: Interupted time series The impact of public transportation strikes on use of a bicycle share program in London: Interrupted time series design Fuller D, Sahlqvist S, Cummins S, Ogilvie D. The impact of public transportation strikes on use of a bicycle share program in London: Interrupted time series design. Preventive Medicine. 2011;54(1):74â 76. Analysis set up Set working directory and importing the data into R options(scipen = 1, digits = 2) opts_chunk$set(warning = FALSE, message = FALSE, error = FALSE) setwd("/users/dogleg/dropbox/conferences/2014 ALR/Workshop Data/") boris_data <- read.csv(" Installing necessary packages install.packages("ggplot2", repos = " library(ggplot2) install.packages("car", repos = " library(car) Objectives We want to examine the impact of 2 tube (read subway) strikes in London on the use of the Boris Bike program. The strikes happened on September 6th, 2010 and October 4th, First we need to generate the appropriate variables for the analysis. First, we need 2 dummy variables that indicate before and after each strike. Creating necessary variables Creating dichotomous variables to assess the immediate impact of the tube strike Key dates that we need for the analysis are the tube strikes 1. September 6, October 4, 2010 The first strike happened at time 39 (September 6, 2010) so we need to create a variable that is 0 before time 39, 1 between time 39 and 65, and 0 again after time 65. boris_data$strike1 <- 1 boris_data$strike1[boris_data$time < 39] <- "0" boris_data$strike1[boris_data$time > 66] <- "0" boris_data$strike1 <- as.numeric(boris_data$strike1) The second strike happened at time 65 (October 4, 2010) so we need to create a variable that is 1 before time 65 and 0 after after time 65.

2 boris_data$strike2 <- boris_data$time - 66 boris_data$strike2 <- ifelse(boris_data$time < 67, c("0"), c("1")) boris_data$strike2 <- as.numeric(boris_data$strike2) Creating continuous variables to assess the change in impact over time Creating a variable that is 0 before the first ban and linear after the first ban boris_data$slope2 <- boris_data$time - 38 boris_data$slope2[boris_data$time < 39] <- "0" boris_data$slope2 <- as.numeric(boris_data$slope2) Creating a variable that is 0 before the second ban and linear after the second ban boris_data$slope3 <- boris_data$time - 66 boris_data$slope3[boris_data$time < 67] <- "0" boris_data$slope3 <- as.numeric(boris_data$slope3) Creating a categorical variable that defines each strike period (i.e., pre, strike1 and strike 2). This is mostly to simply our life when graphing. boris_data$strike[boris_data$time < 39] <- "0" boris_data$strike[boris_data$time > 38 & boris_data$time < 67] <- "1" boris_data$strike[boris_data$time > 66] <- "2" Dataset structure Your dataset should look like below. Notice that the dummy variables change at the correct times. library(xtable) cut_data <- boris_data[30:75, ] print(xtable(cut_data), type = "html") start_date duration_sec t_trip duration_min time time1 time2 t0_int t1_int t2_int strike1 strike2 slope2 slope3 strike Aug Aug Aug Aug Sep Sep Sep Sep Sep Sep Sep Sep Sep Sep Sep Sep Sep Sep

3 48 15-Sep Sep Sep Sep Sep Sep Sep Sep Sep Sep Sep Sep Sep Sep Sep Sep Oct Oct Oct Oct Oct Oct Oct Oct Oct Oct Oct Oct Some simple scatter plots analysis Scatter plot of the Date (x-axis) and Total number of trips (y-axis). Make sure the data is sorted by Date. library(ggplot2) ggplot(boris_data, aes(x = time, y = t_trip)) + geom_point(shape = 1) + xlab("days") + ylab("number of Boris Bike Trips") + theme_bw()

4 Scatter plot of the Date (x-axis) and Total number of trips (y-axis) adding the linear fit (blue) and dashed red lines to represent the two strikes. ggplot(boris_data, aes(x = time, y = t_trip)) + geom_point(shape = 1) + stat_smooth(method = "lm", se = FALSE) + geom_vline(xintercept = 37, colour = "red", linetype = "longdash") + geom_vline(xintercept = 66, colour = "red", linetype = "longdash") + xlab("days") + ylab("number of Boris Bike Trips") + theme_bw() Regression of each of the newly created variables on the total number of trips Regression equation

5 y^ = β 0 + β 1 Tim e 1 + β 2 Strike β 3 Slope β 4 Strike β 5 Slope ε trip_reg <- lm(t_trip ~ time + strike1 + slope2 + strike2 + slope3, data = boris_data) print(xtable(trip_reg), type = "html") Estimate Std. Error t value Pr(> t ) (Intercept) time strike slope strike slope Interpretation of the coefficients is as follows: Beta0 (Intercept) â Value of dependent variable at baseline Beta1 (time1) - Trend prior to 1st intervention implementation Beta2 (strike1) - Difference between the last point prior and first point post the 1st intervention implementation Beta3 (slope2) â Change in trend from pre to post 1st implementation Beta4 (strike2) - Difference between last point in 1st implementation period and first point in 2nd implementation period Beta5 (slope3) â Change in trend from 1st to 2nd implementation periods Getting predicted values from the regression and mergning them to the "boris_data"" data set trip_hat <- fitted(trip_reg) # predicted values boris_data$trip_hat <- fitted(trip_reg) Getting residual values from the regression and mergning them to the "boris_data"" data set trip_resid <- residuals(trip_reg) # residuals boris_data$trip_resid <- (trip_resid) Graph the regression plot ggplot(boris_data, aes(x = time, y = t_trip)) + geom_point(shape = 1) + geom_smooth(aes(group = strike), method = "lm", se = FALSE) + geom_vline(xintercept = 37, colour = "red", linetype = "longdash") + geom_vline(xintercept = 66, colour = "red", linetype = "longdash") + xlab("days") + ylab("number of Boris Bike Trips") + theme_bw()

6 Notes. The plot is a new linear model for each strike period. It is mostly for descriptive purposes. Including CI's in this graph is incorrect as they will be based on the actual model but rather the model used in the graph. That's why I've set the "se=false." Model fit diagnostics Fitted residuals, Normal Q-Q plot opar <- par(mfrow = c(2, 2), oma = c(0, 0, 1.1, 0)) plot(trip_reg, las = 1) The Durbin-Watson statistic is a simple numerical method for checking serial dependence.

7 If the Durbinâ Watson statistic is substantially less than 2, there is evidence of positive serial correlation. As a rough rule of thumb, if Durbinâ Watson is less than 1.0, there may be cause for alarm. Small values of d indicate successive error terms are, on average, close in value to one another, or positively correlated.wiki library(car) durbinwatsontest(trip_reg) ## lag Autocorrelation D-W Statistic p-value ## ## Alternative hypothesis: rho!= 0 Additional notes 1. We could easily add non linear terms (e.g., quadratic) to the models by squaring or cubing the time, slope2 and slope 3 variables. This might help with model fit. 2. My guess is that there is likely a weekend/weekday autocorrelation. We could add a weekend/weekday dummy variable, which would likely improve model fit. 3. If serial autocorrelation continus to be a problem (after adding weekend/weekday variable) we could add Auto-Regressive Integrated Moving Average (ARIMA) components to the model. 4. There is a package designed for interrupted time series regression in R. I have purposefully not shown this package because it is better to understand the model running the analysis the "manual" way. install.packages("segmented", repos = " library(segmented) The end